4400 Higher November 2006 Paper 4 Q25
25
Diagram NOT accurately drawn
The diagram shows one disc with centre \(A\) and radius 4 cm and another disc with centre \(B\) and radius \(x\) cm.
The two discs fit exactly into a rectangular box 10 cm long and 9 cm wide.
The two discs touch at \(P\).
\(APB\) is a straight line.
(a) Use Pythagoras’ Theorem to show that \(x^2 - 30x + 45 = 0\) (4)
(b) Find the value of \(x\).
Give your value correct to 3 significant figures. (3)
Give your value correct to 3 significant figures. (3)
| Scheme | Marks |
|---|---|
| B2 | |
| \((5 - x)^2 + (6 - x)^2 = (x + 4)^2\) | B1 |
| \(25 - 10x + x^2 + 36 - 12x + x^2 = x^2 + 8x + 16\) | B1 |
| (4) |
Notes
B2: two of \((5 - x)\), \((6 - x)\), \((x + 4)\) seen or equiv, eg \((10 - x - 4)\)
B1 for one of these
B1: correct equn not expanded
B1: correct equn expanded
| Scheme | Marks |
|---|---|
| \(\dfrac{30 \pm \sqrt{30^2 - 4 \times 45}}{2}\) | M1 |
| \(\dfrac{30 \pm \sqrt{720}}{2}\) or 28.4 & 1.584 | A1 |
| 1.58 | A1 |
| (3) | |
| (7 marks) |
Notes
M1: Allow \(-30^2\)
The printed scheme shows “Total 4 marks” for this question; the parts add to 7, as on the question paper.